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Theorem xlt2add 9912
Description: Extended real version of lt2add 8433. Note that ltleadd 8434, which has weaker assumptions, is not true for the extended reals (since  0  + +oo  <  1  + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
Assertion
Ref Expression
xlt2add  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A  <  C  /\  B  <  D )  ->  ( A +e B )  <  ( C +e D ) ) )

Proof of Theorem xlt2add
StepHypRef Expression
1 xaddcl 9892 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A +e B )  e.  RR* )
213ad2ant1 1020 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  e.  RR* )
32adantr 276 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  e.  RR* )
4 simp1l 1023 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  e.  RR* )
5 simp2r 1026 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  e.  RR* )
6 xaddcl 9892 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  D  e.  RR* )  ->  ( A +e D )  e.  RR* )
74, 5, 6syl2anc 411 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e D )  e.  RR* )
87adantr 276 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e D )  e.  RR* )
9 xaddcl 9892 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  D  e.  RR* )  ->  ( C +e D )  e.  RR* )
1093ad2ant2 1021 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  e.  RR* )
1110adantr 276 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( C +e D )  e.  RR* )
12 simp3r 1028 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  <  D )
1312adantr 276 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  <  D )
14 simp1r 1024 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  e.  RR* )
1514adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  B  e.  RR* )
165adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR* )
17 simprl 529 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR )
18 xltadd2 9909 . . . . . . . 8  |-  ( ( B  e.  RR*  /\  D  e.  RR*  /\  A  e.  RR )  ->  ( B  <  D  <->  ( A +e B )  <  ( A +e D ) ) )
1915, 16, 17, 18syl3anc 1249 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( B  <  D  <->  ( A +e B )  < 
( A +e
D ) ) )
2013, 19mpbid 147 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  <  ( A +e D ) )
21 simp3l 1027 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  <  C )
2221adantr 276 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  <  C )
234adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  A  e.  RR* )
24 simp2l 1025 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  e.  RR* )
2524adantr 276 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  C  e.  RR* )
26 simprr 531 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  D  e.  RR )
27 xltadd1 9908 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  C  e.  RR*  /\  D  e.  RR )  ->  ( A  <  C  <->  ( A +e D )  <  ( C +e D ) ) )
2823, 25, 26, 27syl3anc 1249 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A  <  C  <->  ( A +e D )  < 
( C +e
D ) ) )
2922, 28mpbid 147 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e D )  <  ( C +e D ) )
303, 8, 11, 20, 29xrlttrd 9841 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  ( A  e.  RR  /\  D  e.  RR ) )  ->  ( A +e B )  <  ( C +e D ) )
3130anassrs 400 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  e.  RR )  ->  ( A +e B )  < 
( C +e
D ) )
32 pnfxr 8041 . . . . . . . . . . . 12  |- +oo  e.  RR*
3332a1i 9 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> +oo  e.  RR* )
34 pnfge 9821 . . . . . . . . . . . 12  |-  ( C  e.  RR*  ->  C  <_ +oo )
3524, 34syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  <_ +oo )
364, 24, 33, 21, 35xrltletrd 9843 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  < +oo )
37 npnflt 9847 . . . . . . . . . . 11  |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
)
384, 37syl 14 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  < +oo  <->  A  =/= +oo )
)
3936, 38mpbid 147 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  A  =/= +oo )
40 pnfge 9821 . . . . . . . . . . . 12  |-  ( D  e.  RR*  ->  D  <_ +oo )
415, 40syl 14 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  <_ +oo )
4214, 5, 33, 12, 41xrltletrd 9843 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  < +oo )
43 npnflt 9847 . . . . . . . . . . 11  |-  ( B  e.  RR*  ->  ( B  < +oo  <->  B  =/= +oo )
)
4414, 43syl 14 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( B  < +oo  <->  B  =/= +oo )
)
4542, 44mpbid 147 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  B  =/= +oo )
46 xaddnepnf 9890 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  ( B  e.  RR*  /\  B  =/= +oo )
)  ->  ( A +e B )  =/= +oo )
474, 39, 14, 45, 46syl22anc 1250 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  =/= +oo )
48 npnflt 9847 . . . . . . . . 9  |-  ( ( A +e B )  e.  RR*  ->  ( ( A +e
B )  < +oo  <->  ( A +e B )  =/= +oo ) )
492, 48syl 14 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( ( A +e B )  < +oo  <->  ( A +e B )  =/= +oo ) )
5047, 49mpbird 167 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  < +oo )
5150adantr 276 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  < +oo )
52 oveq2 5905 . . . . . . 7  |-  ( D  = +oo  ->  ( C +e D )  =  ( C +e +oo ) )
53 mnfxr 8045 . . . . . . . . . . 11  |- -oo  e.  RR*
5453a1i 9 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  e.  RR* )
55 mnfle 9824 . . . . . . . . . . 11  |-  ( A  e.  RR*  -> -oo  <_  A )
564, 55syl 14 . . . . . . . . . 10  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  A
)
5754, 4, 24, 56, 21xrlelttrd 9842 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  C
)
58 nmnfgt 9850 . . . . . . . . . 10  |-  ( C  e.  RR*  ->  ( -oo  <  C  <->  C  =/= -oo )
)
5924, 58syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  C  <->  C  =/= -oo )
)
6057, 59mpbid 147 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  C  =/= -oo )
61 xaddpnf1 9878 . . . . . . . 8  |-  ( ( C  e.  RR*  /\  C  =/= -oo )  ->  ( C +e +oo )  = +oo )
6224, 60, 61syl2anc 411 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e +oo )  = +oo )
6352, 62sylan9eqr 2244 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( C +e
D )  = +oo )
6451, 63breqtrrd 4046 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
6564adantlr 477 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  = +oo )  ->  ( A +e B )  < 
( C +e
D ) )
66 simpr 110 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  D  = -oo )
67 mnfle 9824 . . . . . . . . . 10  |-  ( B  e.  RR*  -> -oo  <_  B )
6814, 67syl 14 . . . . . . . . 9  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <_  B
)
6954, 14, 5, 68, 12xrlelttrd 9842 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  D
)
70 nmnfgt 9850 . . . . . . . . 9  |-  ( D  e.  RR*  ->  ( -oo  <  D  <->  D  =/= -oo )
)
715, 70syl 14 . . . . . . . 8  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  D  <->  D  =/= -oo )
)
7269, 71mpbid 147 . . . . . . 7  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  D  =/= -oo )
7372adantr 276 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  D  =/= -oo )
7466, 73pm2.21ddne 2443 . . . . 5  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  D  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
7574adantlr 477 . . . 4  |-  ( ( ( ( ( A  e.  RR*  /\  B  e. 
RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  /\  D  = -oo )  ->  ( A +e B )  < 
( C +e
D ) )
76 elxr 9808 . . . . . 6  |-  ( D  e.  RR*  <->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
775, 76sylib 122 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
7877adantr 276 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  ->  ( D  e.  RR  \/  D  = +oo  \/  D  = -oo ) )
7931, 65, 75, 78mpjao3dan 1318 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  e.  RR )  ->  ( A +e
B )  <  ( C +e D ) )
80 simpr 110 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  A  = +oo )
8139adantr 276 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  A  =/= +oo )
8280, 81pm2.21ddne 2443 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = +oo )  ->  ( A +e
B )  <  ( C +e D ) )
83 oveq1 5904 . . . . 5  |-  ( A  = -oo  ->  ( A +e B )  =  ( -oo +e B ) )
84 xaddmnf2 9881 . . . . . 6  |-  ( ( B  e.  RR*  /\  B  =/= +oo )  ->  ( -oo +e B )  = -oo )
8514, 45, 84syl2anc 411 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo +e B )  = -oo )
8683, 85sylan9eqr 2244 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  = -oo )
87 xaddnemnf 9889 . . . . . . 7  |-  ( ( ( C  e.  RR*  /\  C  =/= -oo )  /\  ( D  e.  RR*  /\  D  =/= -oo )
)  ->  ( C +e D )  =/= -oo )
8824, 60, 5, 72, 87syl22anc 1250 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( C +e D )  =/= -oo )
89 nmnfgt 9850 . . . . . . 7  |-  ( ( C +e D )  e.  RR*  ->  ( -oo  <  ( C +e D )  <-> 
( C +e
D )  =/= -oo ) )
9010, 89syl 14 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( -oo  <  ( C +e
D )  <->  ( C +e D )  =/= -oo ) )
9188, 90mpbird 167 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  -> -oo  <  ( C +e D ) )
9291adantr 276 . . . 4  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  -> -oo  <  ( C +e D ) )
9386, 92eqbrtrd 4040 . . 3  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e. 
RR* )  /\  ( A  <  C  /\  B  <  D ) )  /\  A  = -oo )  ->  ( A +e
B )  <  ( C +e D ) )
94 elxr 9808 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
954, 94sylib 122 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
9679, 82, 93, 95mpjao3dan 1318 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )  /\  ( A  <  C  /\  B  <  D ) )  ->  ( A +e B )  <  ( C +e D ) )
97963expia 1207 1  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( C  e.  RR*  /\  D  e.  RR* )
)  ->  ( ( A  <  C  /\  B  <  D )  ->  ( A +e B )  <  ( C +e D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ w3o 979    /\ w3a 980    = wceq 1364    e. wcel 2160    =/= wne 2360   class class class wbr 4018  (class class class)co 5897   RRcr 7841   +oocpnf 8020   -oocmnf 8021   RR*cxr 8022    < clt 8023    <_ cle 8024   +ecxad 9802
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-i2m1 7947  ax-0id 7950  ax-rnegex 7951  ax-pre-ltirr 7954  ax-pre-ltwlin 7955  ax-pre-lttrn 7956  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-id 4311  df-po 4314  df-iso 4315  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-fv 5243  df-ov 5900  df-oprab 5901  df-mpo 5902  df-1st 6166  df-2nd 6167  df-pnf 8025  df-mnf 8026  df-xr 8027  df-ltxr 8028  df-le 8029  df-xadd 9805
This theorem is referenced by:  bldisj  14378
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